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# If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?

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If x^1+x^(−1)=5, what is the value of x^4+x^(−4)? [#permalink]

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27 Mar 2017, 03:54
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If $$x^1 + x^{(−1)} = 5$$, what is the value of $$x^4 + x^{(−4)}$$?

A. 527
B. 546
C. 579
D. 600
E. 625
[Reveal] Spoiler: OA

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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)? [#permalink]

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27 Mar 2017, 04:26
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Bunuel wrote:
If x^1 + x^(−1) = 5, what is the value of x^4 + x^(−4)?

A. 527
B. 546
C. 579
D. 600
E. 625

$$x^1 + x^{(−1)} = 5$$ ----------------- I

$$(x+1/x)^2 = x^2 + 1/x^2 + 2$$

$$5^2 = x^2 + 1/x^2 + 2$$ (by using I)

$$x^2 + 1/x^2 = 23$$ --------------- II

$$(x^2+1/x^2)^2 = x^4 + 1/x^4 + 2$$

$$23^2 = x^4 + 1/x^4 + 2$$ (by using II)

$$x^4 + 1/x^4 = 529 - 2 = 527$$

Hence option A is correct
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)? [#permalink]

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27 Mar 2017, 11:16
Bunuel wrote:
If x^1 + x^(−1) = 5, what is the value of x^4 + x^(−4)?

A. 527
B. 546
C. 579
D. 600
E. 625

As explained in the above post, the answer will be A.

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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)? [#permalink]

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29 Mar 2017, 10:10
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Bunuel wrote:
If x^1 + x^(−1) = 5, what is the value of x^4 + x^(−4)?

A. 527
B. 546
C. 579
D. 600
E. 625

We are given that x^1 + x^(−1) = 5, i.e., x + 1/x = 5. We need to determine the value of x^4 + x^(-4), i.e., x^4 + 1/x^4.

Let’s square both sides of the equation x + 1/x = 5.:

(x + 1/x)^2 = 5^2

x^2 + 2(x)(1/x) + 1/x^2 = 25

x^2 + 2 + 1/x^2 = 25

x^2 + 1/x^2 = 23

Now let’s square the above equation:

(x^2 + 1/x^2)^2 = 23^2

x^4 + 2(x^2)(1/x^2) + 1/x^4 = 529

x^4 + 2 + 1/x^4 = 529

x^4 + 1/x^4 = 527

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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)? [#permalink]

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13 Sep 2017, 04:05

Direct attempts to solve for x in this problem will run into quadratics that don't factor and horrible non-integers that need to be raised to fourth powers. Instead, let's focus on manipulating the equation to solve for x^4 + x^−4 directly.

Given the similar structure of the given information, it seems reasonable to begin by squaring the equation x^1+x^−1=5. Be careful, though, not to simply square each term; exponents do not distribute over addition. Instead, recognize the special quadratic. We're looking at two terms added and then squared, so this expression fits the form (a+b)^2=a^2+2ab+b^2. Thus our result will be

(x^1+x^−1^2=5^2

(x^1)^2+2(x^1)(x^−1)+(x^−1)^2=25

x^2+2+x^−2=25

x^2+x^−2=23

Now just repeat the process of squaring both sides once more:

(x^2+x^−2)^2=23^2

x^4+2(x^2)(x^−2)+x^−4=23^2

x^4+2+x^−4=23^2

x^4+x^−4=23^2−2

And it's not even really necessary to calculate 23^2 (which turns out to be 529). 23^2 must end in a 9, so 23^2−2 must end in a 7, and the answer has to be A.
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?   [#permalink] 13 Sep 2017, 04:05
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# If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?

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